Electrons diffraction in crystals

[mendes]

Hello everyone,


It's said that the first Brilloin zone is the k-space where electrons are not diffracted. Electrons with k-vectors on the surface of BZ are diffracted.

How can we understand in terms of electron particules the behavior of electrons associated waves when these waves interfere construtively or destructively ?

 

[asheg]

Hello everyone,
It's said that the first Brilloin zone is the k-space where electrons are not diffracted. Electrons with k-vectors on the surface of BZ are diffracted.
How can we understand in terms of electron particules the behavior of electrons associated waves when these waves interfere construtively or destructively ?

the point is that the
Energy = function(k) = function(k + pi/a)

which mean it should be a periodic function. when you draw energy from k=0 and k=pi/a they reach at k=pi/2a (BZ surface). in this intersection you should add the effect of both. dE/dk = momentum is one side positive and one side negative so the superposition of both are zero.

one can describe situation from wave point of view. when the k vector is on the surface of BZ, the energy of two waves are equal and their k-vector are opposite so you would have a standing wave (total momentum = 0 or dE/dk=0).

 

[asheg]

I get an email that you replied but I can't see it now!

E = f(k) is absolutely periodic because if it isn't so we should draw it all over K-space not in the first BZ and since it's periodic we just draw it in first BZ and in other zones it would be same.

Why? because in a perfect crystal and without perturbations and phonons the minimum wavelength (-> Max k-vector) is the distance with two adjacent atoms which is the surface of first BZ in k-space. but what happens if we increase the k more, the wavelength would be smaller than the distance between atoms but if you plot such wave at atoms it exactly have same values at atoms with a wave having k' = k - pi/a.
So everything in a perfect crystal should remain same with the change k -> k - pi/a

 

[mendes]

I get an email that you replied but I can't see it now!

E = f(k) is absolutely periodic because if it isn't so we should draw it all over K-space not in the first BZ and since it's periodic we just draw it in first BZ and in other zones it would be same.

Why? because in a perfect crystal and without perturbations and phonons the minimum wavelength (-> Max k-vector) is the distance with two adjacent atoms which is the surface of first BZ in k-space. but what happens if we increase the k more, the wavelength would be smaller than the distance between atoms but if you plot such wave at atoms it exactly have same values at atoms with a wave having k' = k - pi/a.
So everything in a perfect crystal should remain same with the change k -> k - pi/a

Thanks a lot for the answers.

I am trying to understand how the values of k pointing on the BZ surface correspond to forbidden values of the energy, and if the semiconductor bandgap can be understood also in terms of wave vectors, k, pointing to a BZ surface.

 

[asheg]

In 1D when k pointing on the BZ surface (k=pi/a) then you will have 2 different possible stationary answers. First, sine-like wave. Second, cosine-like wave. In first situation wave function dominates between atom cores and in second it dominates on the atom cores. One represent the lower point in E-k diagram and the other to the upper point.

You can find a very good description here:
Chapter 7, "Introduction to solid state physics", by Charles Kittle

 

[mendes]

In 1D when k pointing on the BZ surface (k=pi/a) then you will have 2 different possible stationary answers. First, sine-like wave. Second, cosine-like wave. In first situation wave function dominates between atom cores and in second it dominates on the atom cores. One represent the lower point in E-k diagram and the other to the upper point.

You can find a very good description here:
Chapter 7, "Introduction to solid state physics", by Charles Kittle

Thanks a lot for the answers.

Does this mean that sine-like waves describe the conductance electrons while the cosine-like waves describe the valence electrons ?

Electrons corresponding to k values between -pi/a and pi/a, c-a-d inside the the 1st BZ, are supposed to not get reflected and can move freely in the crystal, but since their energy is lower than those descrived by the cosine-like waves, this would mean for me that the valence band electrons can move freely in the crystal (in case what I am saying about the sin-like and cosine-like waves is true) ! A problem ! The valence electrons should not be free !

 

[asheg]

The sine-like wave and cosine-like wave are both stationary. the sine-wave like state dominates between atoms. when the wavefunction's amplitude (Psi^2 which can be described semi classically as electron density and quantum mechanically as probability) is high between atoms the energy is lower (in Electrostatic point of view) compared to cosine-wave like where the density is high on the atoms.
All the states on the surface of BZ are stationary because they are sum of wave and reflected wave and so don't move.
But even non-stationary states do not make current alone because always there is a state with opposite wave vector which compensate it unless electric field be applied.
In some insulators you have valence electrons which can move all over the crystal. Because the electron wavefunction spreads all over the crystal. but sum of all such random movements are zero. When a energy band is full of electrons and there is a energy gap, if you apply electric field it doesn't cause electron current because all the states in the band is full. It's like a full stadium when one wants to move from one side to other side. if it be full of spectators it's very hard to move but if it be half full it would be very easy.
I recommend to take a look at (at least to the pictures)
Chapter 7, "Introduction to solid state physics", by Charles Kittle

 

[mendes]

The sine-like wave and cosine-like wave are both stationary. the sine-wave like state dominates between atoms. when the wavefunction's amplitude (Psi^2 which can be described semi classically as electron density and quantum mechanically as probability) is high between atoms the energy is lower (in Electrostatic point of view) compared to cosine-wave like where the density is high on the atoms.
All the states on the surface of BZ are stationary because they are sum of wave and reflected wave and so don't move.
But even non-stationary states do not make current alone because always there is a state with opposite wave vector which compensate it unless electric field be applied.
In some insulators you have valence electrons which can move all over the crystal. Because the electron wavefunction spreads all over the crystal. but sum of all such random movements are zero. When a energy band is full of electrons and there is a energy gap, if you apply electric field it doesn't cause electron current because all the states in the band is full. It's like a full stadium when one wants to move from one side to other side. if it be full of spectators it's very hard to move but if it be half full it would be very easy.
I recommend to take a look at (at least to the pictures)
Chapter 7, "Introduction to solid state physics", by Charles Kittle

Thanks for your patience :)

Yes, I am looking to the Kittel.

I think my question now is : do the waves, inside the BZ, who then are not diffracted, do they correspond to free electrons that can move freely in the lattice ? The waves outside the Bz are not diffracted either and therefore are similar from this point of view to those inside the BZ, and according to understanding they should correspond to free electrons also, that will move freely in the crystal, that is "conduction electron", then where are the valence electrons ? :)

 

[asheg]

0- You're welcome.

1- When you deal with the perfect crystal without perturbations you can ignore other zones of k-space other than first BZ since k -> k-pi/a

2- The point is that the conduction band differs from valence band in respect to energy. This means that any k-point in k-space has several energies (wave with same wavelength and different frequencies) associated with that point. the conduction band has greater energy compared to valence band but they are in same region of k-space.

3- The electrons in valence band and conduction band both can move. but in the valence band since all of the band is full for any hk (quantum momentum) there is a -hk (since it's full) so you have no net motion. But in the conduction band because it's partially full so when you apply voltage, the equilibrium between these compensating movements (hk and -hk) diminish and there will be waves which are not compensated so there will exist a current. You need a hallf-filled band or gap-less band diagram for a good conductor and if you have a full band (with energy gap to next band) it will not conduct good.